Thanks to Igor Pak for pointing out to us that NP-hardness of the POSITIVE PLANAR 1-IN-3-SAT problem (Proposition 3.3) has apparently been shown already by P. Laroche, Planar 1-in-3 satisfiability is NP-complete, ASMICS Workshop on Tilings, Deuxiemes journees polyominos et pavages, Ecole Normale Superieure de Lyon, 1992, according to a reference in C. Moore and J. M. Robson, Hard tiling problems with simple tiles, Discrete & Computational Geometry 26 (2001), pp. 573-590.